Subgroups — #28374431

Order 13952.

354 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20295563	C₂	1	1	yes	maximal, characteristic, complemented
2	#20295555	C₂	1	1	yes	maximal, characteristic, complemented
3	#20295335	C₂	1	1	yes	maximal, characteristic, complemented
4	#20295333	C₂	1	1	yes	maximal, characteristic, complemented
5	#20295328	C₂	1	1	yes	maximal, characteristic, complemented
6	#20295327	C₂	1	1	yes	maximal, characteristic, complemented
7	#20295281	C₂	1	1	yes	maximal, characteristic, complemented
8	#20295278	C₂	1	1	yes	maximal, characteristic
9	#20295275	C₂	1	1	yes	maximal, characteristic, complemented
10	#20295269	C₂	1	1	yes	maximal, characteristic, complemented
11	#20295250	C₂	1	1	yes	maximal, characteristic, complemented
12	#20295243	C₂	1	1	yes	maximal, characteristic, complemented
13	#20295238	C₂	1	1	yes	maximal, characteristic, complemented
14	#20295048	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₁₀₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13182016	C₂²	1	1	yes	characteristic
17	C₂×#13182014	C₂²	1	1	yes	characteristic
18	C₂×#13182012	C₂²	1	1	yes	characteristic
19	C₂×#13182012	C₂²	1	1	yes	characteristic
20	#16812694	—	2	1	no	—
21	#16812692	—	2	1	no	—
22	#16812691	C₂²	1	1	yes	characteristic
23	#16812690	C₂²	1	1	yes	characteristic
24	#16812688	C₂²	1	1	yes	characteristic
25	#16812684	—	2	1	no	—
26	#16812681	—	2	1	no	—
27	#16812675	C₂²	1	1	yes	characteristic
28	#16812675	C₂²	1	1	yes	characteristic
29	#16812674	C₂²	1	1	yes	characteristic
30	#16812670	C₂²	1	1	yes	characteristic
31	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
32	C₄×D₄₃₆	—	2	1	no	—
33	C₄×D₄₃₆	C₂²	1	1	yes	characteristic
34	D₄×Dic₁₀₉	C₂²	1	1	yes	characteristic
35	#16812659	—	2	1	no	—
36	#16812656	—	2	1	no	—
37	#16812655	—	2	1	no	—
38	C₄²×D₁₀₉	C₂²	1	1	yes	characteristic
39	#16812633	C₂²	1	1	yes	characteristic
40	#16812632	C₂²	1	1	yes	characteristic
41	#16812631	C₂²	1	1	yes	characteristic
42	#16812630	C₂²	1	1	yes	characteristic
43	C₄⋊#11724784	C₂²	1	1	yes	characteristic
44	#16812628	C₂²	1	1	yes	characteristic
45	#16812628	C₂²	1	1	yes	characteristic
46	#16812623	C₂²	1	1	yes	characteristic
47	#16812623	C₂²	1	1	yes	characteristic
48	#16812623	C₂²	1	1	yes	characteristic
49	#16812621	C₂²	1	1	yes	characteristic
50	#16812621	C₂²	1	1	yes	characteristic
51	#16812620	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₁₀₉	C₂²	1	1	yes	characteristic
54	#124×C₁₀₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₁₀₉	C₂²	1	1	yes	characteristic
57	#103×C₁₀₉	C₂²	1	1	yes	characteristic
58	#102×C₁₀₉	C₂²	1	1	yes	characteristic
59	C₂×D₄₃₆	—	2	1	no	—
60	C₂×D₄₃₆	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₀₉	—	2	1	no	—
62	C₂²×Dic₁₀₉	—	2	1	no	—
63	C₂×C₄×D₁₀₉	—	2	1	no	—
64	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₁₀₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₁₀₉	C₂³	1	1	yes	characteristic
67	#13182016	—	4	1	no	—
68	#13182014	—	4	1	no	—
69	#13182012	—	4	1	no	—
70	#13182012	—	4	1	no	—
71	C₂×#11724784	C₂³	1	1	yes	characteristic
72	C₂×#11724784	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₁₀₉	—	2	1	no	—
74	C₄⋊Dic₁₀₉	—	2	1	no	—
75	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₁₀₉	C₂³	1	1	yes	characteristic
78	#13182008	—	2	1	no	—
79	#13182008	—	2	1	no	—
80	#13182008	—	2	1	no	—
81	#13182008	—	2	1	no	—
82	#13182008	—	2	1	no	—
83	C₂²⋊Dic₁₀₉	—	2	1	no	—
84	C₂²⋊Dic₁₀₉	—	2	1	no	—
85	#13182006	—	2	1	no	—
86	#13182006	—	2	1	no	—
87	#13182006	—	2	1	no	—
88	#13182006	—	2	1	no	—
89	#13182006	—	2	1	no	—
90	#13182006	—	2	1	no	—
91	C₄×Dic₁₀₉	—	2	1	no	—
92	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
93	C₄×Dic₁₀₉	D₄	1	1	yes	characteristic
94	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₁₀₉	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₁₀₉	C₂³	1	1	yes	characteristic
98	Q₁₆×C₁₀₉	—	4	1	no	—
99	SD₁₆×C₁₀₉	—	4	1	no	—
100	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₁₀₉	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₁₀₉	—	2	1	no	—
103	C₄⋊C₄×C₁₀₉	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
105	C₂²⋊C₄×C₁₀₉	—	2	1	no	—
106	C₄²×C₁₀₉	—	2	1	no	—
107	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
108	C₂³×C₁₀₉	—	2	1	no	—
109	C₂²×D₁₀₉	—	2	1	no	—
110	C₂²×D₁₀₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₁₀₉	—	2	1	no	—
112	Q₈×C₁₀₉	—	2	1	no	—
113	Q₈×C₁₀₉	—	2	1	no	—
114	Q₈×C₁₀₉	—	2	1	no	—
115	D₄×C₁₀₉	—	2	1	no	—
116	D₄×C₁₀₉	—	2	1	no	—
117	C₂×C₄×C₁₀₉	—	2	1	no	—
118	C₂×C₄×C₁₀₉	—	2	1	no	—
119	C₂×C₄×C₁₀₉	—	2	1	no	—
120	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₁₀₉	—	4	2	no	—
124	C₂×Dic₁₀₉	—	2	1	no	—
125	C₂×Dic₁₀₉	—	2	1	no	—
126	C₂×Dic₁₀₉	—	2	1	no	—
127	C₂×Dic₁₀₉	—	4	2	no	—
128	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₁₀₉	C₂×D₄	1	1	yes	characteristic
131	D₄₃₆	—	2	1	no	—
132	D₄₃₆	—	2	1	no	—
133	D₄₃₆	—	4	2	no	—
134	C₄×D₁₀₉	—	2	1	no	—
135	C₄×D₁₀₉	—	2	1	no	—
136	C₄×D₁₀₉	—	4	1	no	—
137	C₄×D₁₀₉	—	4	1	no	—
138	C₄×D₁₀₉	—	4	2	no	—
139	C₈×C₁₀₉	—	2	1	no	—
140	C₈×C₁₀₉	—	2	1	no	—
141	#11724784	—	2	1	no	—
142	#11724784	—	2	1	no	—
143	C₂²×C₁₀₉	—	4	1	no	—
144	C₂²×C₁₀₉	—	4	1	no	—
145	C₂²×C₁₀₉	—	4	1	no	—
146	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₁₀₉	—	2	1	no	—
148	C₂×D₁₀₉	—	2	1	no	—
149	C₂×D₁₀₉	—	2	1	no	—
150	C₂×D₁₀₉	—	4	1	no	—
151	C₂×D₁₀₉	—	4	1	no	—
152	C₂×D₁₀₉	—	4	1	no	—
153	C₄×C₁₀₉	—	2	1	no	—
154	C₄×C₁₀₉	—	2	1	no	—
155	C₄×C₁₀₉	—	4	1	no	—
156	C₄×C₁₀₉	—	4	1	no	—
157	C₄×C₁₀₉	—	4	1	no	—
158	C₄×C₁₀₉	H(5,𝔽₂)	1	1	yes	characteristic
159	C₄×C₁₀₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₁₀₉	—	2	1	no	—
161	Dic₁₀₉	—	2	1	no	—
162	Dic₁₀₉	—	2	1	no	—
163	Dic₁₀₉	—	4	1	no	—
164	Dic₁₀₉	—	4	1	no	—
165	Dic₁₀₉	—	4	1	no	—
166	C₂×C₁₀₉	—	8	1	no	—
167	C₂×C₁₀₉	#537	1	1	yes	characteristic
168	C₂×C₁₀₉	#575	1	1	yes	characteristic
169	C₂×C₁₀₉	#575	1	1	yes	characteristic
170	D₁₀₉	—	4	1	no	—
171	D₁₀₉	—	8	1	no	—
172	#3256	—	109	1	no	maximal, Sylow
173	C₁₀₉	#3256	1	1	yes	characteristic, complemented, Sylow
174	#529	—	109	1	no	—
175	#529	—	109	1	no	—
176	#479	—	109	1	no	—
177	#479	—	109	1	no	—
178	#478	—	109	1	no	—
179	#478	—	109	1	no	—
180	#470	—	109	1	no	—
181	#469	—	109	1	no	—
182	#468	—	109	1	no	—
183	#468	—	109	1	no	—
184	#464	—	109	1	no	—
185	#464	D₁₀₉	1	1	yes	characteristic, complemented
186	#463	—	109	1	no	—
187	#463	—	109	1	no	—
188	#423	—	109	1	no	—
189	C₂×Q₁₆	C₂×D₁₀₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	109	1	no	—
191	C₂×SD₁₆	C₂×D₁₀₉	1	1	yes	characteristic
192	C₂×D₈	—	109	1	no	—
193	C₂×M₁₆	—	109	1	no	—
194	C₂×M₁₆	—	109	1	no	—
195	#126	—	218	1	no	—
196	#126	—	218	1	no	—
197	#125	—	109	1	no	—
198	#125	—	109	1	no	—
199	#124	—	109	1	no	—
200	#124	C₂×D₁₀₉	1	1	yes	characteristic
201	#123	—	218	1	no	—
202	#123	—	218	1	no	—
203	C₂²⋊Q₈	—	109	1	no	—
204	C₂²⋊Q₈	—	109	1	no	—
205	#121	—	109	1	no	—
206	#121	—	109	1	no	—
207	C₄×Q₈	—	109	1	no	—
208	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
209	C₄×D₄	—	109	1	no	—
210	C₄×D₄	—	109	1	no	—
211	C₄×D₄	—	218	1	no	—
212	C₄×D₄	—	218	1	no	—
213	#117	—	218	1	no	—
214	#117	—	218	1	no	—
215	C₂×C₄²	—	109	1	no	—
216	#107	—	109	1	no	—
217	#107	—	109	1	no	—
218	#106	—	109	1	no	—
219	#106	—	109	1	no	—
220	C₄⋊C₈	—	109	1	no	—
221	C₄⋊C₈	—	109	1	no	—
222	C₄⋊C₈	—	109	1	no	—
223	C₄⋊C₈	C₂×D₁₀₉	1	1	yes	characteristic
224	#103	—	109	1	no	—
225	#103	—	109	1	no	—
226	#103	—	109	1	no	—
227	#103	C₂×D₁₀₉	1	1	yes	characteristic
228	#102	—	109	1	no	—
229	#102	—	109	1	no	—
230	#102	—	109	1	no	—
231	#102	C₂×D₁₀₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
233	C₂×Q₈	C₂²×D₁₀₉	1	1	yes	characteristic
234	C₂×D₄	—	109	1	no	—
235	C₂×D₄	—	218	1	no	—
236	C₂×D₄	—	218	1	no	—
237	C₂×D₄	C₂²×D₁₀₉	1	1	yes	characteristic
238	C₂²×C₄	—	109	1	no	—
239	C₂²×C₄	—	109	1	no	—
240	C₂²×C₄	—	109	1	no	—
241	C₂²×C₄	—	218	1	no	—
242	C₂²×C₄	—	218	1	no	—
243	Q₁₆	—	4	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	436	1	no	—
246	D₈	—	436	1	no	—
247	M₁₆	—	436	1	no	—
248	M₁₆	—	436	1	no	—
249	C₂×C₈	—	109	1	no	—
250	C₂×C₈	—	109	1	no	—
251	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₁₀₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	109	1	no	—
255	C₄⋊C₄	—	109	1	no	—
256	C₄⋊C₄	—	109	1	no	—
257	C₄⋊C₄	—	218	1	no	—
258	C₄⋊C₄	—	218	1	no	—
259	C₄⋊C₄	—	218	1	no	—
260	C₄⋊C₄	—	218	1	no	—
261	C₄⋊C₄	—	218	1	no	—
262	C₄⋊C₄	—	218	1	no	—
263	C₄⋊C₄	—	218	1	no	—
264	C₄⋊C₄	C₂²×D₁₀₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	218	1	no	—
268	C₂²⋊C₄	—	218	1	no	—
269	C₂²⋊C₄	—	218	1	no	—
270	C₂²⋊C₄	—	218	1	no	—
271	C₂²⋊C₄	—	218	1	no	—
272	C₂²⋊C₄	—	218	1	no	—
273	C₂²⋊C₄	—	218	1	no	—
274	C₂²⋊C₄	—	218	1	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	109	1	no	—
277	C₄²	—	109	1	no	—
278	C₄²	—	109	1	no	—
279	C₄²	—	218	1	no	—
280	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	109	1	no	—
283	C₂³	—	218	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	436	2	no	—
291	D₄	—	436	2	no	—
292	D₄	—	218	1	no	—
293	D₄	—	218	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	109	1	no	—
298	C₂×C₄	—	109	1	no	—
299	C₂×C₄	—	109	1	no	—
300	C₂×C₄	—	436	2	no	—
301	C₂×C₄	—	436	1	no	—
302	C₂×C₄	—	436	1	no	—
303	C₂×C₄	—	436	2	no	—
304	C₂×C₄	—	218	1	no	—
305	C₂×C₄	—	218	1	no	—
306	C₂×C₄	—	218	1	no	—
307	C₂×C₄	—	218	1	no	—
308	C₂×C₄	—	218	1	no	—
309	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
310	C₂×C₄	D₄×D₁₀₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	218	1	no	—
315	C₈	—	218	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	436	1	no	—
320	C₂²	—	436	1	no	—
321	C₂²	—	436	1	no	—
322	C₂²	—	218	1	no	—
323	C₂²	—	218	1	no	—
324	C₂²	—	218	1	no	—
325	C₂²	C₂×D₄×D₁₀₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	436	1	no	—
332	C₄	—	436	1	no	—
333	C₄	—	436	1	no	—
334	C₄	—	218	1	no	—
335	C₄	—	218	1	no	—
336	C₄	—	218	1	no	—
337	C₄	#16812744	1	1	yes	characteristic
338	C₄	#16812746	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	872	1	no	—
341	C₂	—	436	1	no	—
342	C₂	#20295605	1	1	yes	characteristic
343	C₂	#20295823	1	1	yes	characteristic
344	C₂	#20295825	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#464×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	#20295278	C₂
hypercenter	#464	D₁₀₉
hypocenter	C₁₀₉	#3256
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #20295563	C₂	semidirect
2. #20295555	C₂	semidirect
3. #20295335	C₂	semidirect
4. #20295333	C₂	semidirect
5. #20295328	C₂	semidirect
6. #20295327	C₂	semidirect
7. #20295281	C₂	semidirect
9. #20295275	C₂	semidirect
10. #20295269	C₂	semidirect
11. #20295250	C₂	semidirect
12. #20295243	C₂	semidirect
13. #20295238	C₂	semidirect
14. #20295048	C₂	semidirect
15. #464×C₁₀₉	C₂	semidirect
173. C₁₀₉	#3256	semidirect
185. #464	D₁₀₉	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20295563	2
#20295555	2
#20295335	2
#20295333	2
#20295328	2
#20295327	2
#20295281	2
#20295278	2
#20295275	2
#20295269	2
#20295250	2
#20295243	2
#20295238	2
#20295048	2
#464×C₁₀₉	2
#3256	109